So being new to the climbing scene I have seen guys taping elbows, arms, and fingers. Can someone explain the reasons begind this, the actual effectiveness, and the benefits. I have heard everything from assisting a muscle tweak, to looking after tendons. Thanks for the input guys.

Avoid taping cos it means you're injured, yes. But once the injury has happened, taping is a must, I reckon, second only to rest. Starting too hard on plastic last year had my tendons in a bad way. I stopped for a while and have since been nursing them back, taping heavily on fingers that were sore, weening them off as they recovered. In addition to the tape, I've also been paying much more attention to my grip, using open crimps as often as possible on plastic and rock. The combination has really, really helped. I've still been able to climb, even with tender pulleys, just watching what I climb and how. So tape is an excellent thing! Insert the usual proviso about obviously not climbing if your fingers/tendons are really hurt here.

I've not heard it used as a remedy for muscle injuries, only tendon support.

There are some pretty cool taping techniques for fingers, elbows, shoulders, back and for most other limbs and body parts. Some are there to stabilize a joint and provide support, others to off-load tendons and muscles, still others help facilitate muscle action and strength while others help increase blood-flow to an area thus helping with pain relief and healing. There are even some which can aid recovery, or so they say...
In climbing most taping techniques are used to support tendons and off-load either them directly or support the little pulleys which make sure the tendon can produce a maximum force in the desired direction.
Prevention is better than cure so make sure you warm up well, stretch correctly (active stretch) and climb smartly...

Generally:
1 Tape on the hands and sometimes fingers is often used to protect the skin when crack climbing (hand taping) or climbing sharp incut holds (fingers). Tape can also help prevent callousing during extended gym sessions (callouses are scar tissue which is weak and rips open easily =flappers=OW!)
2 Taping the fingers, wrists and elbows is done to protect weak or injured tendons. Im not convinced elbow taping works at all, at least it didnt seem to help me. Only use tape for fingers and wrists when recovering strength after injury or when pushing much harder grades than usual on tweaky small holds.

Habitual taping is, as Justin points out, a bad idea. The fingers lose strength and stability and become dependant on the tape.

From a physio point of view Mark has the principals of taping waxed of unloading, facilitating, supporting the muscles and tendons etc... well done one thing to add is the taping also gives the surrounding muscles and joints the proprioreceptive (joint position sense) input which decreases when a muscle or tendon is injured. It can be very effective if applied correctly, but long term use is not advised, as it can ultimately compromised the strength of the structures involved. If you are needing to use taping for an injury long term better you seek further help. Hope that helps, happy climbing!